Cohesive toposes

Bill Lawvere, Categories of spaces may not be generalized spaces, as exemplified by directed graphs , preprint, State University of New York at Buffalo, (1986) Reprints in Theory and Applications of Categories, No. 9, 2005, pp. 1–7 (pdf)

Under the name categories of cohesion these axioms, slightly refined, are presented in

Connections

The notion of connection on a bundle and its various generalizations and variants is fundamental and accordingly the relevant literature is vast, ranging from standard monographs to recent developments. We try to give commented lists of those references that in one way or other connect to our development.

In terms of parallel transport

There are many equivalent statements of the ordinary definition of a connection on a bundle. The following lists references related to the statement that the connection is equivalently encoded in terms of its parallel transport.

Apparently one of the oldest occurrences of the idea that a principal bundleP→XP \to X with connection∇\nabla may be reconstructed from its holonomies around all smooth loops for any fixed base point in the connected base space XX appears in

that generalizes these ideas from loops to general paths. These authors introduced the idea of sitting instants of paths and noticed that the most elegant way to (re)state the maximal equivalence relation on paths which is respected by parallel transport is in terms of thin homotopy.

Barrett originally had something very similar but slightly different. With Caetano and Picken’s relation, the space of thin homotopy classes of paths in XX becomes an groupoid P1(X)\mathbf{P}_1(X) – the path groupoid – internal to diffeological space.

(I am grateful to Christian Fleischhack and to Laurent Guillopé for help with tracking down some of the above links.)

A note on how a 1-form is encoded in the parallel transport that it induces along paths is also in appendix B of

Motivated by the original results by Barrett et al. it was later observed that similarly an abelian gerbe with connection on a simply connected space is entirely encoded in the parallel surface transport that it induces on spheres:

John Baez noticed that these facts suggest that the proper formulation of bundles and higher bundles with connection should be in terms of smooth parallel transport n-functors Pn→A\mathbf{P}_n \to A that smoothly send the path n-groupoid of a space XX to a smooth n-groupoid.

At that point the motivation for this very natural definition was mainly formal, while the relevance of higher nonabelian parallel transport for physics was felt to be compelling but remained somewhat unclarified: